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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring ’11 : Mathematics for Computer Science April 8 Prof. Albert R Meyer revised Wednesday 13 th April, 2011, 14:49 Problem Set 8 Due : April 15 Reading: Chapter 15 – 15.9 , Counting Rules Problem 1. Let X and Y be finite sets. (a) How many binary relations from X to Y are there? (b) Define a bijection between the set OEX ! Y Ł of all total functions from X to Y and the set Y j X j . (Recall Y n is the cartesian product of Y with itself n times.) Based on that, what is j OEX ! Y Ł j ? (c) Using the previous part how many functions , not necessarily total, are there from X to Y ? How does the fraction of functions vs. total functions grow as the size of X grows? Is it O.1/ , O. j X j / , O.2 j X j / ,...? (d) Show a bijection between the powerset, P .X/ , and the set OEX ! f 0;1 g Ł of 0-1-valued total functions on X . (e) Let X WWD f 1;2;:::;n g . In this problem we count how many bijections there are from X to itself. Consider the set B X;X of all bijections from set X to set X . Show a bijection from B X;X to the set of all permuations of X (as defined in the notes). Using that, count B X;X . Problem 2. In this problem, all graphs will have vertices OE1;nŁ WWDf 1;2;:::;n g ; equivalently, all binary relations are on this set OE1;nŁ ....
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- Spring '11
- Computer Science, Binary relation, Type theory, Bijection